Full wavefield inversion (FWI) in exploration seismic processing relies on the calculation of the gradient of an objective function with respect to the subsurface model parameters [12]. An objective function E is usually given as an L2 norm as
                              E          =                                    1              2                        ⁢                          ∫                              ∫                                  ∫                                                                                                                                                                  p                            ⁡                                                          (                                                                                                r                                  g                                                                ,                                                                                                      r                                    s                                                                    ;                                  t                                                                                            )                                                                                -                                                                                    p                              b                                                        ⁡                                                          (                                                                                                r                                  g                                                                ,                                                                                                      r                                    s                                                                    ;                                  t                                                                                            )                                                                                                                                                  2                                        ⁢                                          ⅆ                      t                                        ⁢                                          ⅆ                                              S                        g                                                              ⁢                                          ⅆ                                              S                        s                                                                                                                                ,                            (        1        )            where p and pb are the measured pressure, i.e. seismic amplitude, and the modeled pressure in the background subsurface model at the receiver location rg for a shot located at rs. In iterative inversion processes, the background medium is typically the medium resulting from the previous inversion cycle. In non-iterative inversion processes or migrations, the background medium is typically derived using conventional seismic processing techniques such as migration velocity analysis. The objective function is integrated over all time t, and the surfaces Sg and Ss that are defined by the spread of the receivers and the shots. We define Kd(r)=K(r)−Kb(r) and ρd(r)=ρ(r)−ρb(r), where K(r) and ρ(r) are the true bulk modulus and density, and Kb(r) and ρb(r) are the bulk modulus and the density of the background model at the subsurface location r. We also define the difference between the measured and the modeled pressure to be pd(rg,rs;t)=p (rg,rs;t)−pb(rg,rs;t).
The measured pressure p, satisfies the wave equation
                                              ⁢                                                            ρ                ⁢                                                                  ⁢                                  ∇                                      ·                                          (                                                                        1                          ρ                                                ⁢                                                  ∇                          p                                                                    )                                                                                  -                                                ρ                  K                                ⁢                                  p                  ¨                                                      =                                          -                                  q                  ⁡                                      (                    t                    )                                                              ⁢                              δ                ⁡                                  (                                      r                    -                                          r                      s                                                        )                                                              ,                                          ⁢                                          ⁢          or                                    (        2        )                                                      (                                          ρ                b                            +                              ρ                d                                      )                    ∇                ⁣                                                            ·                                  (                                                            1                                                                        ρ                          b                                                +                                                  ρ                          d                                                                                      ⁢                                          ∇                                              (                                                                              ρ                            b                                                    +                                                      ρ                            d                                                                          )                                                                              )                                            -                                                                                          ρ                      b                                        +                                          ρ                      d                                                                                                  K                      b                                        +                                          K                      d                                                                      ⁢                                  (                                                                                    ρ                        ¨                                            b                                        +                                                                  ρ                        ¨                                            d                                                        )                                                      =                                          -                                  q                  ⁡                                      (                    t                    )                                                              ⁢                              δ                ⁡                                  (                                      r                    -                                          r                      s                                                        )                                                              ,                                    (        3        )            where q(t) is the source signature. By expanding the perturbation terms and keeping only the 1st order Born approximation terms, one can derive the Born scattering equation for the pressure pd,
                                                                        ρ                b                            ⁢                              ∇                                  ·                                      (                                                                  1                                                  ρ                          b                                                                    ⁢                                              ∇                                                  p                          d                                                                                      )                                                                        -                                                            ρ                  b                                                  K                  b                                            ⁢                                                p                  ¨                                d                                              =                      -                          [                                                                                                                  ρ                        b                                            ⁢                                              K                        d                                                                                    K                      b                      2                                                        ⁢                                                            p                      ¨                                        b                                                  -                                                      ρ                    b                                    ⁢                                      ∇                                          ·                                              (                                                                                                            ρ                              d                                                                                      ρ                              b                              2                                                                                ⁢                                                      ∇                                                          p                              b                                                                                                      )                                                                                                        ]                                      ,                            (        4        )            and so pd satisfies
                                                        p              d                        ⁡                          (                                                                    r                    g                                    ⁢                                      r                    s                                                  ;                t                            )                                =                      ∫                                          [                                                                                                    ρ                        b                                            ⁡                                              (                                                  r                          ′                                                )                                                              ⁢                                                                                            K                          d                                                ⁡                                                  (                                                      r                            ′                                                    )                                                                                                                      K                          b                          2                                                ⁡                                                  (                                                      r                            ′                                                    )                                                                                      ⁢                                                                                            p                          ¨                                                b                                            ⁡                                              (                                                                              r                            ′                                                    ,                                                                                    r                              s                                                        ;                            t                                                                          )                                                                              -                                                                                    ρ                        b                                            ⁡                                              (                                                  r                          ′                                                )                                                              ⁢                                          ∇                                              ·                                                  (                                                                                                                                                      ρ                                  d                                                                ⁡                                                                  (                                                                      r                                    ′                                                                    )                                                                                                                                                              ρ                                  b                                  2                                                                ⁡                                                                  (                                                                      r                                    ′                                                                    )                                                                                                                      ⁢                                                          ∇                                                                                                p                                  b                                                                ⁡                                                                  (                                                                                                            r                                      ′                                                                        ,                                                                                                                  r                                        s                                                                            ;                                      t                                                                                                        )                                                                                                                                              )                                                                                                                    ]                            *                                                g                  b                                ⁡                                  (                                                            r                      g                                        ,                                                                  r                        ′                                            ;                      t                                                        )                                            ⁢                              ⅆ                                  V                  ′                                                                    ,                            (        5        )            where V′ is the volume spanned by r′, and gb is the Green's function in the background medium.
One can derive the equations for the gradients of pb using Eq. (5) and by considering the fractional change δpb due to fractional change δKb and δρb over an infinitesimal volume dV,
                                                        ∂                              p                b                                                    ∂                                                K                  b                                ⁡                                  (                  r                  )                                                              =                                                                                          ρ                    b                                    ⁡                                      (                    r                    )                                                  ⁢                dV                                                              K                  b                  2                                ⁡                                  (                  r                  )                                                      ⁢                                          g                b                            ⁡                              (                                                      r                    g                                    ,                                      r                    ;                    t                                                  )                                      *                                                            p                  ¨                                b                            ⁡                              (                                  r                  ,                                                            r                      s                                        ;                    t                                                  )                                                    ,                                  ⁢        and                            (        6        )                                                                    ∂                              p                b                                                    ∂                                                ρ                  b                                ⁡                                  (                  r                  )                                                              =                                    F                              -                1                                      ⁢                          {                                                dV                                                            ρ                      b                                        ⁡                                          (                      r                      )                                                                      ⁢                                                      ∇                                                                  G                        b                                            ⁡                                              (                                                                              r                            g                                                    ,                                                      r                            ;                            f                                                                          )                                                                              ·                                      ∇                                                                  P                        b                                            ⁡                                              (                                                  r                          ,                                                                                    r                              s                                                        ;                            f                                                                          )                                                                                                        }                                      ,                            (        7        )            where Pb=F{x}, Pd=F{pd}, Gb=F{gb}, and F and F−1 are the Fourier transform and the inverse Fourier transform operators.
By using Eqs. 6 and 7, and using the reciprocity relationship ρb(r)Gb(rg,r)=ρb(rg)Gb(r,rg),
                                                                                                              ∂                    E                                                        ∂                                                                  K                        b                                            ⁡                                              (                        r                        )                                                                                            =                                ⁢                                  -                                      ∫                                          ∫                                              ∫                                                                              p                            d                                                    ⁢                                                                                    ∂                                                              p                                b                                                                                                                    ∂                                                                                                K                                  b                                                                ⁡                                                                  (                                  r                                  )                                                                                                                                              ⁢                                                      ⅆ                            t                                                    ⁢                                                      ⅆ                                                          S                              g                                                                                ⁢                                                      ⅆ                                                          S                              s                                                                                                                                                                                                                                                  =                                ⁢                                                      -                                                                                                                        ρ                            b                                                    ⁡                                                      (                            r                            )                                                                          ⁢                        dV                                                                                              K                          b                          2                                                ⁡                                                  (                          r                          )                                                                                                      ⁢                                      ∫                                          ∫                                              ∫                                                                                                            (                                                              i                                ⁢                                                                                                                                  ⁢                                2                                ⁢                                                                                                                                  ⁢                                π                                ⁢                                                                                                                                  ⁢                                f                                                            )                                                        2                                                    ⁢                                                                                    P                              b                                                        ⁡                                                          (                                                              r                                ,                                                                                                      r                                    s                                                                    ;                                  f                                                                                            )                                                                                                                                                                                                                                                                  ⁢                                                                            G                      b                                        ⁡                                          (                                                                        r                          g                                                ,                                                  r                          ;                          f                                                                    )                                                        ⁢                                                            P                      d                      *                                        ⁡                                          (                                                                        r                          g                                                ,                                                                              r                            s                                                    ;                          f                                                                    )                                                        ⁢                                      ⅆ                    f                                    ⁢                                      ⅆ                                          S                      g                                                        ⁢                                      ⅆ                                          S                      s                                                                                                                                              =                                ⁢                                                                                                                              ρ                          b                                                ⁡                                                  (                          r                          )                                                                    ⁢                      dV                                                                                      K                        b                        2                                            ⁡                                              (                        r                        )                                                                              ⁢                                      ∫                                          ∫                                                                                                                                  p                              .                                                        b                                                    ⁡                                                      (                                                          r                              ,                                                                                                r                                  s                                                                ;                                t                                                                                      )                                                                          ⁢                                                  ∫                                                                                                                                                      ρ                                  b                                                                ⁡                                                                  (                                                                      r                                    g                                                                    )                                                                                                                                                              ρ                                  b                                                                ⁡                                                                  (                                  r                                  )                                                                                                                      ⁢                                                                                          g                                b                                                            ⁡                                                              (                                                                  r                                  ,                                                                                                            r                                      g                                                                        ;                                                                          -                                      t                                                                                                                                      )                                                                                      *                                                                                                                                                                                                                                      ⁢                                                                                                                              p                          .                                                d                                            ⁡                                              (                                                                              r                            g                                                    ,                                                                                    r                              s                                                        ;                            t                                                                          )                                                              ⁢                                          ⅆ                                              S                        g                                                              ⁢                                          ⅆ                      t                                        ⁢                                          ⅆ                                              S                        s                                                                              ,                                                                    ⁢                                  ⁢                                  ⁢        and                            (        8        )                                                                                                      ∂                  E                                                  ∂                                                            ρ                      b                                        ⁡                                          (                      r                      )                                                                                  =                            ⁢                              ∫                                  ∫                                      ∫                                                                  p                        d                                            ⁢                                                                        ∂                                                      p                            b                                                                                                    ∂                                                                                    ρ                              b                                                        ⁡                                                          (                              r                              )                                                                                                                          ⁢                                              ⅆ                        t                                            ⁢                                              ⅆ                                                  S                          g                                                                    ⁢                                              ⅆ                                                  S                          s                                                                                                                                                                                            =                            ⁢                                                -                                      dV                                                                  ρ                        b                                            ⁡                                              (                        r                        )                                                                                            ⁢                                  ∫                                      ∫                                          ∫                                                                                                    P                            d                            *                                                    ⁡                                                      (                                                                                          r                                g                                                            ,                                                                                                r                                  s                                                                ;                                f                                                                                      )                                                                          ⁢                                                                              ∇                                                                                          P                                b                                                            ⁡                                                              (                                                                  r                                  ,                                                                                                            r                                      s                                                                        ;                                    f                                                                                                  )                                                                                                              ·                                                                                                                                                                                                          ⁢                                                ∇                                                            G                      b                                        ⁡                                          (                                                                                                    r                            g                                                    ⁢                          r                                                ;                        f                                            )                                                                      ⁢                                  ⅆ                  f                                ⁢                                  ⅆ                                      S                    g                                                  ⁢                                  ⅆ                                      S                    s                                                                                                                          =                            ⁢                                                -                                      dV                                                                  ρ                        b                                            ⁡                                              (                        r                        )                                                                                            ⁢                                  ∫                                      ∫                                                                  ∇                                                                              p                            b                                                    ⁡                                                      (                                                          r                              ,                                                                                                r                                  s                                                                ;                                t                                                                                      )                                                                                              ·                                                                                                                                                            ⁢                                                [                                      ∫                                                                  ∇                                                  (                                                                                                                    g                                b                                                            ⁡                                                              (                                                                  r                                  ,                                                                                                            r                                      g                                                                        ;                                                                          -                                      t                                                                                                                                      )                                                                                      *                                                                                                                            ρ                                  b                                                                ⁡                                                                  (                                                                      r                                    g                                                                    )                                                                                                                                                              ρ                                  b                                                                ⁡                                                                  (                                  r                                  )                                                                                                                      ⁢                                                                                          p                                d                                                            ⁡                                                              (                                                                                                      r                                    g                                                                    ,                                                                                                            r                                      s                                                                        ;                                    t                                                                                                  )                                                                                                              )                                                                    ⁢                                              ⅆ                                                  S                          g                                                                                                      ]                                ⁢                                  ⅆ                  t                                ⁢                                                      ⅆ                                          S                      s                                                        .                                                                                        (        9        )            One can then use Eqs. 8 and 9 to perform full wavefield inversion in an iterative manner.
Reverse-time migration (RTM) is based on techniques similar to gradient computation in FWI, where the forward propagated field is cross-correlated with the time-reversed received field. By doing so, RTM overcomes limitations of ray-based migration techniques such as Kirchhoff migration. In RTM, the migrated image field M at subsurface location r is given asM(r)=∫∫pb(r,rs;t)∫gb(r,rg;−t)*p(rg,rs;t)dSgdtdSs,  (10)which is very similar to the gradient equation 8 of FWI.
While Eqs. 8 and 9 provide the framework for inverting data into subsurface models, the convergence of the inversion process often is very slow. Also, RTM using Eq. 10 suffers weak amplitude in the deep section due to spreading of the wavefield. Many attempts have been made to improve the convergence of FWI or improve the amplitude of the reverse-time migration by using the Hessian of the objective function [9], i.e., a second derivative of the objective function. Computation of the Hessian, however, is not only prohibitively expensive in computational resources, but it requires prohibitively large storage space for a realistic 3-D inversion problem. Furthermore, FWI using the full Hessian matrix may result in suboptimal inversion [2].
One may be able to perform more stable inversion by lumping non-diagonal terms of the Hessian into the diagonal terms [2]. These, however, still require computation of the full Hessian matrix or at least a few off-diagonal terms of the Hessian matrix, which can be costly computations. While one may choose to use the diagonal of the Hessian only [11], this is valid only in the high frequency asymptotic regime with infinite aperture [1, 7].
Plessix and Mulder tried to overcome these difficulties by first computing an approximate diagonal Hessian, then by scaling these by zανpβ, where z is the depth and νp is the compressional wave velocity [7]. From numerical experiment, they have determined that the best scaling parameter is z0.5νp0.5. This approach, however, does not provide quantitative inversion of the subsurface medium parameters with correct units, since only approximate scaling has been applied. Furthermore, this approach was applied to RTM where only variations in compressional wave velocity is considered, and so may not be applicable to FWI where other elastic parameters such as density and shear wave velocity vary in space.